Imbalanced $d$-wave superfluids in the BCS-BEC crossover regime at finite temperatures

Singlet pairing in a Fermi superfluid is frustrated when the amounts of fermions of each pairing partner are unequal. The resulting “imbalanced superfluid” has been realized experimentally for ultracold atomic gases with $s$-wave interactions. Inspired by high-temperature superconductivity, we investigate the case of $d$-wave interactions and find marked differences from the $s$-wave superfluid. Whereas $s$-wave imbalanced Fermi gases tend to phase separate in real space, in a balanced condensate and an imbalanced normal halo, we show that the $d$-wave gas can phase separate in reciprocal space so that imbalance and superfluidity can coexist spatially. We show that the mechanism explaining this property is the creation of polarized excitations in the nodes of the gap. The Sarma mechanism, present only at nonzero temperatures for the $s$-wave case, is still applicable in the temperature zero limit for the $d$-wave case. As a result, the $d$-wave BCS superfluid is more robust with respect to imbalance and a region of the phase diagram can be identified where the $s$-wave BCS superfluidity is suppressed whereas the $d$-wave superfluidity is not. When these results are extended into the Bose–Einstein condensate limit of strongly bound molecules, the symmetry of the order parameter matters less. The effects of fluctuations beyond mean field is taken into account in the calculation of the structure factor and the critical temperature. The poles of the structure factor (corresponding to bound molecular states) are less damped in the $d$-wave case as compared to that in $s$ wave. On the BCS side of the unitarity limit, the critical temperature $T_c$ follows the temperature $T^{*}$ corresponding to the pair binding energy and as such will also be more robust against imbalance. Possible routes for the experimental observation of the $d$-wave superfluidity have been discussed.

Figure 1

Even a small difference between the chemical potential leads to population imbalance. This plot shows the contours of $E_{\mathbf{k}}=\zeta$ for different values of $\zeta$ in the $k_x,k_y$ plane for a $d$-wave interaction characterized by $k_0=k_1=10k_F$. Within the regions $E_{\mathbf{k}}<\zeta$, spin-polarized Bogoliubov excitations are present, which carry the excess spin component of the imbalanced gas.

Figure 2

(Color online) The dependence of the pair binding energy on the interaction strength for the $s$-wave (top) and $d$-wave (bottom) interactions is influenced by the imbalance $\delta n∕n=0.0,\dots ,0.9$. On the BCS side, the imbalance destroys $s$-wave Cooper pairs for all values of $a_s<0$, but fails to break up the $d$-wave Cooper pairs.

Figure 3

(Color online) Critical temperature for the fermion system with the $d$-wave scattering calculated taking into account the fluctuations around the saddle point as a function of the inverse scattering length (red solid dots). The saddle-point critical temperature $T^{*}$ is plotted with the solid black curves.

Figure 4

(Color online) Excitation region of a gas of interacting fermions in $(q,\omega )$ space (the case of the $d$-wave scattering) at $T=T_c$ for $k_0=10$, $k_1=10$, and $\cos \theta =1∕2$. The shaded area shows the continuum of the two-particle excitations. The black curve denotes the lower-frequency bound of the damping area. The dashed red curve shows the points given by $\omega ={\Omega }_b\left(\mathbf{q}\right)$, where ${\Omega }_b\left(\mathbf{q}\right)={\omega }_b\left(\mathbf{q}\right)-2\mu$ with the energy ${\omega }_b\left(\mathbf{q}\right)$ of the two-body bound state.

Figure 5

(Color online) 3D plot of the structure factor $S(q,\omega )$ (a) for the $s$-wave scattering and (b) for the $d$-wave scattering at $T=T_c$ in the weak-coupling regime. The critical temperatures are given in units of the Fermi temperature $T_F\equiv {\hslash }^2{k}_F^2∕2m{k}_B$. At the top, there are the contour density plots for $S(q,\omega )$.

Figure 6

(Color online) 3D plot of the structure factor $S(q,\omega )$ (a) for the $s$-wave scattering and (b) for the $d$-wave scattering at $T=T_c$ in the strong-coupling regime.

Figure 7

(Color online) Integration contour in the complex $z$ plane. The solid dots indicate the poles $z=i{\Omega }_n$, $n=0,\pm 1,\pm 2,\dots$.